From 097f244236b682cc77c6ecf25be4150091d8daf9 Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Sun, 2 Oct 2022 03:04:14 +0000 Subject: CTAN sync 202210020304 --- info/mathtrip/src/calc3.tex | 58 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 58 insertions(+) create mode 100644 info/mathtrip/src/calc3.tex (limited to 'info/mathtrip/src/calc3.tex') diff --git a/info/mathtrip/src/calc3.tex b/info/mathtrip/src/calc3.tex new file mode 100644 index 0000000000..dc314f132e --- /dev/null +++ b/info/mathtrip/src/calc3.tex @@ -0,0 +1,58 @@ +%This command produces the text of the calculus in the first column +%of the first horizontal area +% +%The command has one parameter: +% 1) The width of the mathematical text +\newcommand\TEightCalculusThree[1]{% + %Command to put a strut in these formulae + \def\TEightCalcC{\rule[-5ex plus .5ex minus 1ex]{0pt}{6ex plus 2ex minus 1ex}} + %A much bigger strut + \def\TEightCalcBig{\rule[-5ex plus .5ex minus 1ex]{0pt}{12ex plus 2ex minus 1ex}} + \parbox[t]{#1}{% + \TEightSeriesCalculusFontSize + \begin{DisplayFormulae}{62}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulae}{\BigChar}{\StyleBold} + %Formula 62 + \Fm{\int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \arccos \frac{a}{\vert x\vert}\quad a > 0} + \Fm{\int \frac{dx}{x^2 \sqrt{x^2 \pm a^2}} = \mp \frac{\sqrt{x^2 \pm a^2}}{a^2 x}} + \Fm{\int \frac{\xdx}{\sqrt{x^2 \pm a^2}} = \sqrt{x^2 \pm a^2}} + \Fm{\int \frac{\sqrt{x^2 \pm a^2}}{x^4} \, dx = \mp \frac{(x^2 + a^2)^{3/2}}{3a^2 x^3}} + \Fm{\int \frac{dx}{a x^2 + bx + c} = \left\{ + \begin{array}{lr} + \displaystyle \frac{1}{\sqrt{b^2 -4ac}} \ln \left\vert\frac{2ax + b - \sqrt{b^2 -4ac}}{2ax + b + \sqrt{b^2 -4ac}}\right\vert + &\text{if $b^2 > 4ac$\TEightCalcC} \\ + \displaystyle \frac{2}{\sqrt{4ac - b^2}} \arctan \frac{2ax + b}{\sqrt{4ac - b^2}} + &\text{if $b^2 < 4ac$\TEightCalcC} \\ + \end{array}\right. + } + \Fm{\int \frac{dx}{\sqrt{a x^2 + bx + c}} = \left\{ + \begin{array}{lr} + \displaystyle \frac{1}{\sqrt{a}} \ln \left\vert 2ax + b + 2\sqrt{a} \sqrt{ax^2 + bx + c}\right\vert + &\text{if $a > 0$\TEightCalcC}\\ + \displaystyle \frac{1}{\sqrt{- a}} \arcsin \frac{-2ax - b}{\sqrt{b^2 - 4ac}} + &\text{if $a < 0$\TEightCalcC}\\ + \end{array}\right. + } + \def\FirstPart{\int \sqrt{a x^2 + bx + c} \, dx =} + \FmPartA{\FirstPart} + \FmPartB{\FirstPart}{\frac{2ax + b}{4a} \sqrt{a x^2 + bx + c} + + \frac{4ax - b^2}{8a} \int \frac{dx}{\sqrt{a x^2 + bx + c}}} + \Fm{\int \frac{\xdx}{\sqrt{a x^2 + bx + c}} = \frac{\sqrt{a x^2 + bx + c}}{a} - \frac{b}{2a} + \int \frac{dx}{\sqrt{a x^2 + bx + c}}} + \Fm{\TEightCalcBig\int \frac{dx}{x \sqrt{a x^2 + bx + c}} = \left\{ + \begin{array}{l@{\hspace{.3ex plus .3ex minus .1ex}}r} + \displaystyle \frac{-1}{\sqrt{c}} \ln \left\vert\frac{2\sqrt{c} \sqrt{ax^2 + bx + c} + bx + 2c}{x}\right\vert + &\text{\small if $c > 0$}\\ + \displaystyle \frac{1}{\sqrt{- c}} \arcsin \frac{bx + 2c}{\vert x \vert\sqrt{b^2 - 4ac}} + &\text{\small if $c < 0$} \\ + \end{array}\right. + } + \Fm{\int x^3 \sqrt{x^2 + a^2} \, dx = (\frac{1}{3} x^2 - \frac{2}{15} a^2)(x^2 + a^2)^{3/2}} + %Formula 72 + \Fm{\int x^n \sin (ax) \, dx = - \frac{1}{a} x^n \cos (ax) + \frac{n}{a} \int x^{n-1} \cos (ax) \, dx} + \Fm{\int x^n \cos (ax) \, dx = \frac{1}{a} x^n \sin (ax) - \frac{n}{a} \int x^{n-1} \sin (ax) \, dx} + \Fm{\int x^n e^{ax} \, dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax} \, dx} + \Fm{\int x^n \ln (ax) \, dx = x^{n+1}\left(\frac{\ln (ax)}{n+1} - \frac{1}{(n+1)^2}\right)} + \Fm{\int x^n (\ln ax)^m \, dx = \frac{x^{n+1}}{n+1}(\ln ax)^m - \frac{m}{n+1}\int x^n (\ln ax)^{m-1} \, dx} + \end{DisplayFormulae} + } +} -- cgit v1.2.3